1. In the end of semester examination, 32% students failed in Economics, Mathematics, 46%in Business, 12% in Economics and Mathe
matics, 9% Mathematics and Business, 10% in Economics and Business and 3% in all the three. Using Venn diagram, determine, i. Number of students that failed in exactly one course. ii. Number of students that failed in exactly two courses. iii. Number of students who passed in all the three courses. iv. Number of students who failed in at least one course.
(ii) 3% of students failed all three courses, 12% failed economics and mathematics, 9% in math and business, and 10% in econ and business. In other words, the students in these groups failed at least the two mentioned courses. Then 12% - 3% = 9% failed only econ and math <em>but not</em> business; 9% - 3% = 6% failed math and business <em>but not</em> econ; and 10% - 3% = 7% failed econ and business <em>but not</em> math. These groups are mutually exclusive, so the total percentage of students that failed exactly two courses is (ii) 9% + 6% + 7% = 22%.
32% failed econ, which means 32% - 12% - 10% + 3% = 13% failed <em>only</em> econ. Similarly, 46% failed business, so 46% - 9% - 10% + 3% = 30% failed <em>only</em> business. But we don't know how many students failed math, so we can't determine how many failed <em>only</em> math... And consequently we can't determine the proportion of students that make up the other categories.